On the generalized Carlitz module

TL;DR

This paper explores formal similarities between Euler's gamma function and the Anderson-Thakur function within the framework of the generalized Carlitz module, highlighting common structures and differences in their exponential functions.

Contribution

It introduces the generalized Carlitz module as a unifying framework to compare these two functions and analyzes their associated exponential functions.

Findings

Identifies formal similarities between Gamma and Anderson-Thakur functions

Highlights differences in their exponential functions

Proposes the generalized Carlitz module as a common structural framework

Abstract

The aim of this note is to gather formal similarities between two apparently different functions; {\em Euler's function} $\Gamma$ and {\em Anderson-Thakur function} $\omega$. We discuss these similarities in the framework of the {\em generalized Carlitz's module}, a common structure which can be helpful in framing the theories of both functions. We further analyze several noticeable differences while investigating the {\em exponential functions} associated to these structures.